Conceptual Real Analysis

I have been taking real analysis II this semester and I am starting to get a better grasp over the broad subject of analysis and integration.

However, I feel like my understanding is completely problem-oriented. I tried talking to a colleague of mine about real analysis in a conceptual manner and we couldn't talk for more than just a few minutes. I do have some basic understanding, for example: integration is clearly something similar to the measure of size, or something of that nature. However, I am curious about a more extensive, conceptual discussion of the subject which just focuses on the intuition, motivation, and interpretations of the results.

Does anyone know a good book or good resource where I can approach this?

I am not aware of any such books, but I am not a fan of the "conceptual" approach in general, so take what I'm about to say with a grain of salt. For me, the motivation for real analysis was single-variable calculus. So to think of integration as the "area under the curve," as you're taught, is a wonderful aid to intuition. Similarly, thinking of differentiation as the "slope of the tangent line" works wonders. Real analysis, after all, is just a generalization of single-variable calculus - maybe you work with metric spaces, nowhere differentiable functions, etc., but the intuition and, by extension, the ideas for most proofs come from basic calculus.

So why don't I think there should be a book devoted to fleshing out this intuition? Because most "conceptual" books I see try to convince you that a theorem is true. It seems to me much more efficient (and valuable to the student, in the long run) just to prove that it's true.

That's a good point, and I kind of agree. It is a technical subject, and when you're learning a technical subject, you should be focusing on technical results, like proofs. However, I've been reading a lot of proofs on the subject, and it's not like I'm disdaining the proofs or trying to get around that.

I'm definitely not interested in a "conceptual proof" book that attempts to prove things non-rigorously. I'm just looking for something that explores the interpretation of the results a little bit more. Surely, there is enough depth to the subject that you can discuss the broad meaning of the results more than by just saying "this is the area under the curve". Perhaps it is a bit ambitious to devote a whole book to such interpretations, but, well I really don't know.

Yeah I'm finding measure theory, as well as my other math classes, difficult to conceptualize. It seems I have to throw all my intuition away and start over. I'm hoping the more I mess with stuff, the more comfortable I'll get with it and the better I'll understand it in general. I doubt there's a way to come to a conceptual understanding without getting your hands dirty.